Mechanics Research Communications 37 (2010) 246–255

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Mechanics Research Communications

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On a micromorphic model for the synovial fluid in the human knee

Veturia Chiroiu a,*, Valeria Mosnegut�u a, Ligia Munteanu a, Rodica Ioan b

a Institute of Solid Mechanics, Romanian Academy, Ctin Mille 15, Bucharest 010141, Romaniab Spiru Haret University, Ion Ghica 13, Bucharest 030045, Romania

a r t i c l e i n f o a b s t r a c t

Article history:Received 6 April 2009Received in revised form 25 November 2009Available online 11 December 2009

Keywords:Human kneeSynovial fluidHAPRG4Micromorphic fluidGenetic algorithm

0093-6413/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.mechrescom.2009.12.003

* Corresponding author. Fax: +40 21 3126736.E-mail address: veturiachiroiu@yahoo.com (V. Chi

In this paper, an alternative model for the synovial fluid in the human knee is proposed. The synovial fluidis modeled as a micromorphic fluid containing deformable material particles with 12 degrees of freedom,namely three translations, three rotations and six stretches and shears. Two micromorphic viscosity coef-ficients are introduced as functions of hyaluronan and PRG4/lubricin concentrations and are recon-structed by a genetic algorithm based on experimental data. The model is validated by applying it towell established rheological measurements of synovial fluid, such as shear-thinning (Cooke et al., 1978).

� 2009 Elsevier Ltd. All rights reserved.

3

1. Introduction

The normal knee joint is surrounded by the synovium, a mem-brane that produces a synovial fluid into the joint cavity. This fluidforms a thin microscopic layer of fluid of about 50 at the surface ofcartilage. Normal synovial fluid is clear, colourless, thick andstringy, like eggwhite. The word syn derives from ovum, Latin foregg, because of the egg-like consistency of the synovial fluid. Car-tilage surfaces in synovial fluid support pressures up to 200 atm(Morrell et al., 2005) slide on each other with friction coefficientsin the range 0.0005–0.04 (Forster and Fisher, 1996) and usuallydo not show signs of wear over the entire life of a healthy person.

Healthy synovial fluid contains hyaluronan (hyaluronic acidHA) with concentration of ch = 1–4 (mg/ml), which is a polymerof disaccharides composed of D-glucuronic acid and D-N-acetyl glu-cosamine joined by glycosidic bonds (Weissman and Meyer, 1954;Mazzucco et al., 2004). Hyaluronan is synthesized by the synovialmembrane and secreted into the joint cavity to increase the viscos-ity and elasticity of articular cartilages and lubricate the surfacesbetween synovium and cartilage (Ateshian et al., 1991, Ateshian,1993; Ateshian et al., 2003).

Synovial fluid also contains proteoglycan (PRG4) with concen-tration of cl = 0.05–0.35 (mg/ml) (Schmid et al., 2001; Rhee et al.,2005; Simkin, 2000). PRG4/lubricin is present in synovial fluidand on the surface (superficial layer) of the articular cartilageand therefore plays an important role in joint lubrication (Jay,2004). Lubricin is a large, water soluble glycoprotein encoded by

ll rights reserved.

roiu).

the PRG4 gene. It has a molecular weight of 206 � 10 u (molarmass in g/mol = average molecular mass in u; the most commonunits of molar mass are g/mol since using these units, the numeri-cal value equals the average molecular mass in units of u) andconsists of approximately equal proportions of protein and glyco-saminoglycanbs. Experimental measurements by electron micro-scope show that the lubricin molecule is a partially extendedflexible rod with a fully extended length of l = 200 ± 50 nm and adiameter of a few nanometers (Radin et al., 1970). In solution, itoccupies a smaller spacial domain than would be expected fromstructural predictions (Swann et al., 1981). The composition ofPRG4/lubricin is typical of mucin proteins, which form the mucouscoating of many surfaces in the human body, e.g. teeth, eye-lids,respiratory and gastrointestinal tract, reproductive organs, etc.(Bansil et al., 1995, Zappone et al., 2007). Mucins bind to the epi-thelial surfaces and create a protective layer, in the form of a gel.

The HA and PRG4 are responsible for the boundary-layer lubri-cation, which reduces the friction between the opposite surfaces ofthe cartilage, together with surface active phospholipids (SAPL)with a concentration of at cs = 0.1 (mg/ml) (Schwarz and Hills,1998; Swann et al., 1981). The loss of these functions causes thehuman recessive disorders, such as joint inflammation, redness,pain, swelling, and fluid accumulation. An analysis of the synovialfluid could diagnose the cause of disorders and the precocious jointfailure. However, an important difficulty in the analysis of thesynovial fluid is represented by the unknown viscosity coefficients,which are in general functions of the lubricin’s composition.

While the properties of synovial fluid may affect the tribology ofjoint replacement prostheses, the flow parameters of joint fluidhave not yet been examined in the context of total and revision

Fig. 2.1. (a) Synovial fluid aggregates, and (b) the macromass elements containingaggregates as micromass elements.

V. Chiroiu et al. / Mechanics Research Communications 37 (2010) 246–255 247

knee arthroplasty. The viscous properties of synovial fluid obtainedat total and revision knee arthroplasty varied widely and aredegenerated with respect to synovial fluid from healthy patients(Dowson, 1990).

A comprehensive understanding of viscoelastic properties of thenormal and contaminated synovial fluid has not completely beenundertaken until now. Remarkable results concerning the elasticproperties of the synovial fluid were reported by Ogston andStanier (1953) and the normal stress by Caygill and West (1969).Viscous and elastic moduli at 2.5 Hz, as well as the crossoverfrequency and the modulus in normal synovial fluid, were reportedby Balazs (1982). Compared to normal, all modulus parameterswere markedly decreased in patients undergoing total and revisionknee arthroplasty. Also, the flow properties of joint fluids inpatients undergoing total knee arthroplasty was analysed byMazzucco et al. (2002).

In this paper we present an alternative model for the synovialfluid. The model is validated by applying it to well established rhe-ological measurements of synovial fluid, such as shear-thinning(Cooke et al., 1978).

Instead of the theory of mixture, we base our model on a micro-morphic theory (Eringen 1970a,b, 1998; Cosserat, 1909). The syno-vial fluid is modeled as a micromorphic fluid, that means a fluidwith deformable material particles, which is endowed with twelvedegrees of freedom, namely three translations, three rotations andsix stretch and shears. So, the components of the synovial fluidpossess twelve degrees of freedom: Three for the centroidal motion(macrodeformations), three for the rotation and six for its microde-formations. The motivation consists of exploring the mechanism oflubrication by taking into account the molecular interactions be-tween HA, PRG4 and SAPL in a synovial fluid (Blewis et al., 2007).The difference between the micropolar theory and the classicaltheory of synovial fluids is given by considering the couple stressesand internal micromoment of inertia. In the micropolar theory, themotion of the synovial fluid in the normal walking is described notonly by a deformation, but also by a microrotation responsible forsix degrees of freedom. The interaction of the synovial componentsis transmitted not only by a force, but also by a torque, resultingin asymmetric force stresses and couple stresses. In addition tothe microrotation, the micromorphic theory introduces amacrodeformation.

This paper deals with an inverse approach based on the micro-morphic theory, for the reconstruction of the viscosity coefficientsfrom available experimental data related to the normal walking ofthe human knee.

2. Formulation of the problem

According to the micromorphic theory, the motion of a materialvolume element DV is described in terms of the average motionsof large numbers of aggregates contained in DV. Fig. 2.1a schemat-izes an synovial aggregate as a microvolume element DV(a),a = 1, 2, 3, . . . which consists of different components (with200 nm mean diameter) as Na+, Cl, urea, urate, glucose, lipids,phospholipids, proteins, albumin, c-globulin, hyaluronates, PRG4/lubricin with various concentrations. These components representthe ‘‘deformable material particles” or ‘‘particles” of DV(a). The Pa

0

represents an arbitrary point named a ‘‘material point” of each‘‘particle”.

In the Lagrange system of reference, the position vector of amaterial point in DV(a) is denoted by X(a) = X + R(a), where X isthe position vector of the center of mass P of DV, and R(a) is the rel-ative position vector of a material point of DV(a) from P (Fig. 2.1b).During motion and deformation, in the Euler system of reference,the volume DV0 becomes Dv, the spatial position at time t of the

point P becomes p, and the spatial position of points in Dv(a) arex(a) = x + n(a), where x is the spatial position vector of the centerof mass p of Dv, and n(a) is the relative position vector of a spatialpoint of Dv(a) from p (Fig. 2.1b).

Consider a small volume element DV consisting of deformablematerial particles in an undeformed body. Let the rectangular coor-dinates XK, K = 1, 2, 3 or X denote the position vector of the centerof mass of DV. If DV moves and deforms under external effects, thenew position x(a) of X(a) will be x(a) = x(X, t) + n(a)(X, R, t).

By assuming that the material particles in DV undergo a homo-geneous deformation about x, we have n = vK(X, t)RK, where sum-mation over the repeated index K is understood. The motion andmicromotion of the synovial components are expressed by (Erin-gen, 1970b, Munteanu et al., 2006)

xk ¼ x̂kðX; tÞ; k ¼ 1;2;3; nk ¼ vkKðX; tÞRK ; K

¼ 1;2;3; ð2:1Þ

where the rectangular coordinates of x and n are denoted by xk andnk, respectively. The capital Latin indices refer to the material coor-dinates XK and small indices to the spatial coordinates xk.

The Eq. (2.1)1 describes the macromotion of a synovial fluid. Themicromotion of a synovial fluid is described by (2.1)2. In analogywith the micromorphic theory of the blood (Eringen, 1970b,Munteanu et al., 2006), the micromotion is equivalent to the rota-tion and the microdeformations of the particles. In (2.1)2 vkK(X, t)represent the three deformable directors attached to of the parti-cles so that the microdeformation is the time evolution of thesedeformable directors. Both Eqs. (2.1) are assumed to be uniquelyinvertible so that the single-valued functions XK(x, t) and vKk(x, t),which are the inverses of xK(X, t) and vkK(X, t), respectively, canbe determined from (2.1)

XK ¼ X̂Kðx; tÞ; vkKvKl ¼ dkl; vkKvLk ¼ dKL: ð2:2Þ

The deformation-rate tensors, which characterize the synovialfluid deformation and motion, are essential for the characterizationof the viscosity resistance of synovial fluid, namely

Dxk;K

Dt¼ ~vk;lxl;k;

DvK;k

Dt¼ �XK;l~ml;k; _vkl ¼ mklvlK ; _vKk ¼ �vKlmlk;

ð2:3Þ

where DDt is the material time derivative, the dot means the ordinary

time derivative, indices after a comma denote the partial deriva-tives, ~vkðx; tÞ is the velocity field, and vkl(x, t) the gyration tensor.

248 V. Chiroiu et al. / Mechanics Research Communications 37 (2010) 246–255

Expressions (2.3)2 follow from the definition of the gyrationtensor

mkl ¼ _vkKvKl or _vkK ¼ mklvlK : ð2:4Þ

Deformation rate tensors are fundamental for the characteriza-tion of the synovial fluid. They are given by tensors

akl ¼ ~vk;l � mlk; bklm ¼ vkl;m; ckl ¼12ðvkl þ mlkÞ; ð2:5Þ

The synovial fluid is modeled as an incompressible fluid, so thatwe have a macroincompressibility condition ~vk;k ¼ 0, and a micro-incompressibility condition vkk = 0. The macroincompressibilitydoes not lead to total incompressibility of the synovial fluid, henceboth constraints must be satisfied.

From (2.5) we have

akk ¼ ~vk;k � mkk ¼ 0; ckk ¼ vkk ¼ 0: ð2:6Þ

The constitutive equations for a synovial fluid are obtain fromthe Onsanger postulate (Eringen, 1970b; Mos�negut�u, 2008)

tkl ¼ �pdkl þ Dtkl; pij ¼ �@w@q�1 dij; Dtkl ¼

@U@akl

;

Dmkl ¼@U@blk

; Dskl ¼@U@clk

; qk ¼@U

@ðh;k=hÞ; ð2:7Þ

where tkl is the stress tensor, pij is the micropressure, Dtkl is the dy-namic part of the stress tensor, w = e � hg = w(h, q�1, tr(i)) is theHelmholtz free energy, h is the temperature, g is the entropy, q isthe mass density of the synovial fluid, ikm = imk is the microinertiamoments, U = U(akl, bkl, h,k/h, q�1, ikl) is the dissipative function,mklm is the third-order moment-stress tensor (couple stress tensor),Dmkl is the dynamic part of mklm, skl = slk is the symmetric micro-stress tensor (microstress average), and Dskl is the dynamic part ofskl. The dynamic parts of tkl and skl depend on akl and ckl, whilethe dynamic parts of mklm depend on bkl.

The total tensors tkl, skl and mklm are defined by the sum of thestatic and dynamic terms, i.e. skl = Sskl + Dskl, mklm = Smklm + Dmklm,tkl = Stkl + Dtkl, where the subscripts S and D refer to static and dy-namic quantities, respectively. The static terms are obtained fromthe Helmholtz free energy w, while the dynamic terms are ob-tained from the dissipative function U, which characterizes thedissipative properties of the lubricin and is expressed in terms ofthe deformation rate tensors akl, bkl and ckl given by (2.5). In the fol-lowing, for describing the synovial deformation and motion wesuppose that the static parts vanish.

In order to obtain the constitutive equations of a synovial fluid,consider the dissipation function U

U ¼ �~pðx; tÞakk �-ðx; tÞckk þU1 þU2; ð2:8Þ

where ~p and - are Lagrange multipliers. The dissipation functionsU1, U2 depend on the invariants of the vector and tensor variablesand are given by

2U1 ¼ lðtrða2Þ þ 8trðc2Þ þ 8trðacÞ þ trðaaTÞÞ;

2U2 ¼ sðbijjbikk þ bijjbkik þ bijibjkk þ bijibkjk þ bijkbijkÞ þKh

h;ih;i;

ð2:9Þ

where l(ch, cl) is the viscosity coefficient for the stress tensor Dtkl

and the microstress tensor Dskl, s(ch, cl) is the viscosity coefficientfor the stress moments mklm, and K

h is the heat conduction coeffi-cient. These viscosity coefficients are functions of the concentrationch of HA and the concentration cl of PRG4. The restrictions placed onthese moduli to make U P 0 for all independent variations of akl, bkl

and ckl.Substitution of (2.5) (2.6) (2.8), and (2.9) into (2.7) leads to the

constitutive equations for synovial fluid

Dtkl ¼ �pdkl þ lð~vk;l þ ~v l;k þ vkl þ v lkÞ;Dskl ¼ �-dkl þ 2lð~vk;l þ ~v l;k þ vkl þ v lkÞ;

Dmklm ¼ sðv li;i þ v il;iÞdkm þ sðvmi;i þ v im;iÞdkl; qk ¼Kh

h;k:

ð2:10Þ

The balance laws of a synovial fluid are composed of:

1. Conservation of mass

r � ~v ¼ 0; ð2:11Þ

where ~v is the velocity vector.

2. Conservation of microinertia moments

Dikl

Dt� ikrv lr � ilrvkr ¼ 0; ð2:12Þ

where vkl is the microstrain tensor.

3. Balance of momentum

tkl;k þ q fl �D~v l

Dt

� �¼ 0; ð2:13Þ

where fl is the body force per unit mass.

4. Balance of momentum moments

mklm;k þ tml � sml þ qðllm � rlmÞ ¼ 0; ð2:14Þ

where llm are body moments per unit mass, and rlm is the spininertia.

5. Balance of energy

�q _eþ tklakl þ sklckl þmklmblmk þ qk;k þ qh ¼ 0; ð2:15Þ

where e is the internal energy density per unit mass, qk is the heatvector directed outward the body, and h is the heat source per unitmass.

6. The entropy inequality

�qð _wþ g _hÞ þ tklakl þ sklckl þmklmblmk þ1h

qkh;k P 0: ð2:16Þ

By substituting the constitutive equations into the balance laws,we obtain the equations of motion for the synovial fluid

r � ~v ¼ 0;Dikl

Dt� ikrv lr � ilrvkr ¼ 0;

� ~p;l þ lð~v l;kk þ vkl;k þ v lk;kÞ þ q fl �D~v l

Dt

� �¼ 0;

pml þ ð-� ~pÞdml þ sðvmk;lk þ vkl;km þ vmk;kl þ vkm;kl þ v lm;kkÞ

þ lð~vm;l � ~v l;m � vml � v lmÞ þ q llm � irmDv lr

Dt� v lkvkr

� �� �¼ 0;

� qcvDhDt� h

@pkl

@hvkl þ Dtklakl þmklmblmk þ Dsklckl þ qk;k þ qh ¼ 0:

ð2:17Þ

Equations (2.17) represent 20 partial differential equations forthe determination of 20 unknown functions ~vk; ikl;vkl; ~p and -.

Now, we reduce the number of equations by considering theflow of the synovial fluid in a channel in two-dimensions consist-ing of two parallel walls located at �h < x2 < h from the centerlinealong the x1-axis with constant temperature. We have

V. Chiroiu et al. / Mechanics Research Communications 37 (2010) 246–255 249

~v2 ¼ 0; ~v1 ¼ vðx2Þ; mkl ¼ mklðx2Þ; m11 ¼ m22 ¼ 0: ð2:18Þ

The adherence boundary conditions are given by

v ¼ 0; mkl ¼ 0 at x2 ¼ �h: ð2:19Þ

Body loads are assumed to vanish. The flow of the synovial fluidis assumed to be steady and incompressible, with zero microinertiamoments. We see that the continuity equation (2.17)1 is satisfiedwith (2.18). From (2.17)3 and (2.17)4 we obtain 6 equations of mo-tion for 6 unknowns ~p;p11;p22;v12; m21 and ~v , namely

� ~p;1 þ lð~v ;22 þ v21;2 þ m12;2Þ ¼ 0;� ~p;2 þ 2lv22;2 ¼ 0;p11 þ-� ~pþ sv11;22 � 2lv11 ¼ 0;p22 þ-� ~pþ 5sv22;22 � 2lv22 ¼ 0;p12 þ sð2v12;22 þ v21;22Þ � lðv12 þ v21 þ ~v ;2Þ ¼ 0;p12 þ sðv12;22 þ 2v21;22Þ � lðv12 þ v21 þ ~v ;2Þ ¼ 0;

ð2:20Þ

It is possible to reduce again the number of equations of motion(2.20) to 3 by some assumptions and algebra. To do this, we seethat the dissipation functions in (2.9) become

U1 ¼ lðaðijÞaðijÞ þ 4cijcij þ 4aðijÞcijÞ;2U2 ¼ sðm2

11;2 þ 5m222;2 þ 2m2

12;2 þ 2m221;2 þ 2m12;2m21;2Þ;

ð2:21Þ

where a(ij) is the symmetric part and a[ij] the antisymetric part of aij.In the view of simplicity we neglect a[ij].

The dissipation functions are nonnegative if l P 0 and s > 0.Integration of (2.20)1 and (2.20)2 leads to

lðv ;2 þ v21 þ m12Þ ¼ p0;1x2 þ C1; ~p ¼ 2lv22 þ p0ðx1Þ; ð2:22Þ

where C1 is an integration constant which vanishes becausev,2 = v12 = v21 = 0. From (2.22)2 the pressure ~p can be determinedwhen the applied pressure p0(x1) is given. Substituting p from(2.22)2 into (2.20)3 and (2.20)4 and using the microincompressibil-ity condition v11 + v22 = 0, we obtain

p11 þ-� ~pþ sv11;22 � 2lv11

¼ 0; p22 þ-� ~p� 5sv11;22 þ 2lv11 ¼ 0: ð2:23Þ

By adding and subtracting these equations, we obtain

p11 � p22 þ 6sv11;22 � 4lv11 ¼ 0;12ðp11 þ p22Þ þ-� pþ 2sv11;22 ¼ 0: ð2:24Þ

Assume now that the turbulence level in a synovial fluid is notso far from isotropy, so that p11 = p22 and p12 = 0. From (2.24)2 andthe boundary conditions (2.19) and (2.22)1, we have

2sm12;22 þ sm21;22 � 2lm21 ¼12

p0;1x2; ð2:25Þ

sm12;22 � lm12 þ 2sm21;22 ¼ p0;1x2 ð2:26Þ

Finally, we have three equations of motion given by (2.20)1,(2.25) and (2.26) with three unknowns ~v;v12 and v21. The explicitsolutions for this problem are very difficult to be obtained. Fromour experience in analysing the blood motion in vessels with smallradius (Munteanu et al., 2006), the solutions of the problem(2.20)1, (2.25) and (2.26) are the same as the solutions of the sameproblem but treated in cylindrical coordinates. It is worth mention-ing that by introducing the cylindrical coordinates, the features ofthe film of lubricant’s motion on a surface are not altered. Thestatement that the solutions to the problem (2.20)1, (2.25) and(2.26) and the analytical solutions obtained for the problem writ-ten in cylindrical coordinates are equivalent is based on thenumerical comparison of the results for both these problems.These results indicate a good matching, as we also observed inthe case of the blood flow. Therefore, introducing cylindrical coor-

dinates is not a restrictive assumption; it is only for obtaining theanalytical solutions in an easier manner. Consequently, considerthe synovial fluid flow in a circular vessel with small diameters2h, with z along the axis of the cylinder (x1), and 0 6 r 6 h, r beingin the x2 direction. We denote the velocity of the fluid byw ¼ ~vzðrÞ ¼ ~v (~v r ¼ ~vh ¼ 0). Here mrz(r), mzr(r) (mzz = mrr = 0) are thecomponents of the gyration tension. The equations of motion(2.20)1, (2.25) and (2.26) become

lðw;r þ mrz þ mzrÞ ¼12

p0;zr;

2s mzr;rr þ1rmzr;r �

1r2 mzr

� �þ s v rz;rr þ

1r

v rz;r �2lsþ 1

r2

� �mrz

� �

¼ 12

rp0;z;

s mzr;rr þ1rmzr;r �

lsþ 1

r2

� �mzr

� �þ 2s mrz;rr þ

1rmrz;r �

1r2 mrz

� �

¼ 12

rp0;z;

ð2:27Þ

where p0 = a cos mz + b sin mz, 0 6m 6 1. The constants a, b and mcharacterize the applied pressure p0(z). The general boundary con-ditions are given by

mzrð0Þ ¼ mrzð0Þ ¼ 0;

wðhÞ ¼ 0;12½mzrðhÞ þ mrzðhÞ� ¼ m0

s ðch; clÞ;

12½mrzðhÞ � mzrðhÞ� ¼ m0

aðch; clÞ:

ð2:28Þ

Let us consider, firstly, that the viscosity coefficients are con-stants. For m0

s ðch; clÞ ¼ m0aðch; clÞ ¼ 0, the solutions of (2.27) are

ww0¼ 1� r

h

� �2þ 3

2l 1� r2

h2

� �; ð2:29Þ

mzr ¼ �p0;zh2l

r; ð2:30Þ

mrz ¼p0;zh2l

K1I1ðk1rÞ þ K2I1ðk2rÞ � 12

r� �

; ð2:31Þ

where I1 is the modified bessel function of order 1, and

w0 ¼ �p0;zh

2

4l; K1 ¼ �

2k21

k21 �

2ls

; K2 ¼ �2k2

2

k22 �

2ls

: ð2:32Þ

Here k1 and k2 are two positive roots of the following equation

k2 � 2ls

� �k2 � l

s

� �¼ 0: ð2:33Þ

For m0s ðch; clÞ–0 and m0

aðch; clÞ–0, the general solutions of (2.27)are given by

ww0¼ 1� r

h

� �2þ 2A1ð1þ K1Þ

k1h½I0ðk1rÞ � I0ðk1hÞ�

� 2A2ð1þ K2Þk2h

½I0ðk2rÞ � I0ðk2hÞ� þ 32l 1� r2

h2

� �; ð2:34Þ

mzr ¼p0;zh2l½A1I1ðk1rÞ þ A2I1ðk2rÞ � r�; ð2:35Þ

mrz ¼p0;zh2l

K1I1ðk1rÞ þ K2I1ðk2rÞ � 12

r� �

; ð2:36Þ

where A1 and A2 are determined from the boundary conditions(2.28) as functions of ch and cl, and I0 is the modified Bessel functionof order 0.

250 V. Chiroiu et al. / Mechanics Research Communications 37 (2010) 246–255

Starting with the solutions (2.34) (2.35) (2.36) of the micromor-phic model for the synovial fluid, in what follows we construct thedirect approach and the inverse approach in order to understandthe influence of the modelling assumptions on the obtainedresults.

In the direct approach, the velocity w=w0 ¼ ~vzðrÞ=~vz0 and gyra-tion components mzr and mrz are computed by assuming that the vis-cosity coefficients l and s are known. Referring to the estimationof the viscosity coefficients with respect to the concentrationparameters ch, cl and cs, no information is available in the literature.The aim of the inverse approach is to estimate the dependence of land s on ch, cl and cs, from rheological measurements of thevelocities w/w0 furnished by the caption technique of motion forwalking normal knees (Anybody Technology) (Blewis et al., 2007;Bhusari, 2007; Plaas et al., 2006; Smith and Ghosh, 1987; Hills,2000, Cooke et al., 1978; Ogston and Stanier, 1953; Caygill andWest, 1969; Jay et al., 2007).

3. Direct approach

To validate the solutions 2.34, 2.35 and 2.36, we apply them torheological measurements of the synovial fluid, such as shear-thin-ning (Cooke et al., 1978; Oates et al., 2006). In the direct approach,the values for the viscosity coefficients l and s are assumed to beapproximately known. The properties of the cartilage, synovium,and synoval fluid utilized in the model include the cartilage andsynovium surface area, and the synovial fluid volume. The totalsurface area in the human knee joint is 121 � 10�4m2 for the car-tilage (Eckstein et al., 2001) and 277 � 10�4m2 for the synovium(Davies, 1946), and the synovial fluid volume enclosed by thesesurfaces is 1.1 ml (Ropes et al., 1940). The synovium is2h = 50 lm thick and the cell volume fraction is varying from0.42 to 0.67.

Our first aim is to investigate the dependence of solutions on ch,cl, for given cs = 0.1. For this, we compute from (2.28) the dimen-sionless functions �A1 ¼ A1=A10, �A2 ¼ A2=A20, with m0

s ðch; clÞ ¼m0

aðch; clÞ ¼ 0:5v0, where A10 and A20 correspond tom0

s ðch; clÞ ¼ m0aðch; clÞ ¼ 0. The parameters ch, cl and cs are estimated

from the published values available in the literature for a normalhuman knee, namely cs = 0.1 (mg/ml), ch = 1– 4 (mg/ml) and

Fig. 3.1. Variation of the functions A1 and A2

cl = 0.05–0.35 (mg/ml) (Blewis et al., 2007). As appears in Fig. 3.1,the functions A1 and A2 are continuous with respect to ch, cl.

Once A1 and A2 are known, we can compute the solutions 2.34,2.35 and 2.36 written in a dimensionless form as �w ¼ w

lw0, �v1 ¼ vzr

v0,

�v2 ¼ vrzv0

, with v0, a gyration reference. In Fig. 3.2, these solutionswere calculated for cs = 0.1 (mg/ml), cl = 0.2 (mg/ml), and three val-ues for ch, i.e. ch = 1, ch = 2.5 and ch = 3.5 (mg/ml). In Fig. 3.3, thesolutions 2.34, 2.35 and 2.36 are computed for cs = 0.1 (mg/ml),ch = 2.5 (mg/ml), and three values for cl, i.e. cl = 0.05, cl = 0.2 andcl = 0.35 (mg/ml). s = 2.271. In these plots, l = 3.177, s = 2.271 arechosen. The sensitivity of the solutions on the ch and cl is the keyfor the rheopexy of the synovial fluid, namely a specific behaviorof the fluid during steady shear. The stress in low-viscosity poly-mer solutions under steady shear usually reaches its steady statevalue rapidly. In contrast, the behavior of synovial fluis indicatesthat some type of structure is forming in time. A shear thinningfluid displays the increase of stresses with respect to time duringsteady shear and the decrease of the viscosity with respect toincreasing the shear rate. To understand this property, the stressDt12 and the microstress Ds12 are plotted in Fig. 3.4 as functionsof the deformation rate c12 given by (2.5). As can be seen inFig. 3.4, the stress and microstress grow in time during steadyshear. In Fig. 3.4a, the experimental curve (Oates et al., 2006) isshown to be bounded by two theoretical stresses Dt12, computedfor ch = 1 and cl = 0.05, and ch = 3.5 and cl = 0.2, respectively, as wellas a rate of deformation c12 = 0.05 s�1. The corresponding viscosi-ties for c12 = 0.05 s�1, are l = 0.472, and respectively 4.301, andalso s = 0.692, and respectively 3.251. With increasing shear rateto c12 = 100 s�1, the corresponding viscosities decrease tol = 0.467, and respectively 4.278, and also to s = 0.665, and respec-tively 3.187. For the aforementioned reasons, the results of the di-rect problem are found to be in qualitative agreement with theexperimental predictions (Cooke et al., 1978).

4. Inverse approach

We have introduced two micromorphic viscosity coefficients land s as function of the parameters ch and cl. In the direct approachwe assumed that l and s are approximately known from someavailable experimental data. The aim of the inverse approach is

with respect to the parameters ch and cl.

Fig. 3.2. Solutions �w; �v1 and �v2 for ch = 1, ch = 2.5 and ch = 3.5 (mg/ml).

V. Chiroiu et al. / Mechanics Research Communications 37 (2010) 246–255 251

to evaluate the viscosity coefficients l and s as functions of ch andcl based on rheological measurements of the velocities w/w0.

The problem to be addressed here is the inverse problem. Weseek to determine information about the viscosity coefficients byusing experimental information concerning the solution (2.36)from rheological measurements of the velocities w/w0 furnishedby the caption technique of motion for walking normal knees.

The aim is to use the difference between the experimentalVexp

j ¼ fwðrj; ch; clÞ; j ¼ 1;2; . . . ;mg, and theoretically predictedvelocities Vj ¼ fwðrj; ch; clÞ; j ¼ 1;2; . . . ;mg to provide a procedurewhich iteratively corrects the control parameters towards valuesleading to the lowest difference between predictions and experi-mental observations. The unknown parameters are the viscositycoefficients l and s, which are functions of ch and cl, and piecewisecontinuous on ch min 6 ch 6 ch max and cl min 6 cl 6 cl max.

In the following, we consider that l and s are approximated bypolynomials, characterized by the coefficients cl

k , csk , al

k , ask and bl

k ,bs

k , k = 1, 2

l ¼ cl0 þ al

1 ch þ bl1 cl þ al

2 c2h þ bl

2 c2l ; s ¼ cs

0 þ as1ch þ bs

1cl þ as2c2

h þ bs2c2

l :

ð4:1Þ

To extract the cl0 , cs

0 alk , as

k and blk , bs

k , k = 1, 2, from experimentaldata, an objective function J must be chosen to measure theagreement between theoretical and experimental data. This objec-tive function is built based on three restrictions expressed as threeterms, namely

I ¼ m�1Xm

j¼1

ðVj � Vexpj Þ

2 þ ð4mÞ�1Xm

j¼1

X3

l¼1

d2lj þ d2

4: ð4:2Þ

The first term controls the agreement between the predictedand experimental values, the second term contains the residualsdlj; l ¼ 1;2;3, j ¼ 1;2; . . . ;m, which control the verification of 2.34,2.35 and 2.36) at points rj; j ¼ 1;2; . . . ;m

d1j ¼wðrj; ch; clÞ

w0; d2j ¼ mzrðrj; ch; clÞ ¼ mzrj;

d3j ¼ mrzðrj; ch; clÞ ¼ mrzj; j ¼ 1;2; . . . ;m; ð4:3Þ

and the last term is the residual d4 which controls the fulfillment of(2.30), i.e.

d24 ¼ m2

zrð0Þ þ m2rzð0Þ þw2ðhÞ

þ 12

mzrðhÞ þ mrzðhÞ½ � � m0s ðch; clÞ

� �2

þ 12

mrzðhÞ � mzrðhÞ½ � � m0aðch; clÞ

� �2

: ð4:4Þ

The objective function (4.2) is maximized by using a geneticalgorithm for determining an accurate set of 10 unknowns, so thatthe residuals have the values very close to zero. These residualsevaluate the verification of all governing equations and boundaryconditions and assure the existence and uniqueness of the results(Mihailescu and Chiroiu, 2005; Chiroiu and Chiroiu, 2003).

We define the fitness as F ¼ I0I

, with I0 ¼ m�1Pmj¼1ðV

expj Þ

2. As theconvergence criterion of iterative computations, we use the non-dimensional norm of differences between ? and ?0, namely theexpression Z, defined as Z ¼ 1

2 log10II0

.A binary vector with 10 genes is used to represent the real val-

ues of unknowns. The length of the vector depends on the required

Fig. 3.3. Solutions �w; �v1 and �v2 for cl = 0.05, cl = 0.2 and cl = 0.35 (mg/ml).

Fig. 3.4. Time dependent stress Dt12 and microstress Ds12 in steady shear at 0.05 s�1 for the synovial fluid model.

252 V. Chiroiu et al. / Mechanics Research Communications 37 (2010) 246–255

precision, which in this case is six places after the decimal point.The genetic parameters are assumed to be as follow: number ofpopulations 200, ratio of reproduction 1.0, number of multi-pointcrossover 1, probability of mutation 0.5, and maximum numberof generations 500. We show that the knowledge of the experi-mental data Vexp

j ¼ fwðrj; ch; clÞ; j ¼ 1;2; . . . ;mg is sufficient todetermine the unknowns by simultaneously satisfying as manyrestrictions in (4.2) as possible.

The numerical experiments preformed show that for m < 100,the genetic algorithm has no solution. For m above this value, we

obtain one, two or more solutions. In all of the examples presentedin the next section we considered m = 270. The genetic algorithmpresents four sets of solutions in the case of the objective functionwith the first term only (Fig. 4.1). The solutions are obtained in thiscase after 44 iterations. Three sets of solutions are obtained after94 iterations in the case of the objective function with the firsttwo terms (Fig. 4.1). For the entire objective function expressedin (4.2), the GA provides a unique solution obtained after 124 iter-ations. For this solution, the fitness value shows a better agreementbetween the predicted and experimental values of the control

Fig. 4.1. Number of solutions of the genetic algorithm depending on the restrictionsnumber.

Table 4.1The viscosity coefficient l (10�3 Pa s) for an immobilized normal knee.

ch/cl (mg/ml) 0.05 0.1 0.15 0.2 0.25 0.3 0.35

1 0.413 0.611 1.001 2.306 3.080 3.383 3.8871.5 0.588 0.820 1.272 2.591 3.137 3.474 4.0312 1.023 1.189 2.769 2.823 3.325 3.522 4.1662.5 1.167 1.529 2.994 3.055 3.661 4.053 4.2403 2.045 2.132 3.131 3.278 4.134 4.280 4.3303.5 3.521 3.788 3.977 4.189 4.431 4.798 4.8494 4.613 4.716 4.848 4.979 5.053 5.474 5.606

Table 4.2The viscosity coefficient s (10�3 Pa s) for an immobilized normal knee.

ch/cl (mg/ml) 0.05 0.1 0.15 0.2 0.25 0.3 0.35

1 0.629 0.721 1.010 1.306 1.868 2.163 2.3551.5 0.865 0.944 1.279 1.392 1.981 2.318 2.4732 0.912 1.200 1.774 1.824 2.175 2.471 2.9162.5 1.073 1.539 2.099 2.155 2.355 2.571 3.0963 1.240 1.741 2.437 3.079 3.883 4.169 4.2973.5 2.411 2.501 2.580 3.189 4.030 4.296 4.3464 2.599 2.727 2.851 3.479 4.166 4.301 4.374

Table 4.3The viscosity coefficient l (10�3 Pa s) for a walking knee.

ch/cl (mg/ml) 0.05 0.1 0.15 0.2 0.25 0.3 0.35

1 0.469 0.667 1.111 2.416 3.190 3.493 3.9431.5 0.698 0.930 1.382 2.647 3.193 3.584 4.5912 1.583 1.409 2.935 2.933 3.491 3.634 4.3862.5 1.223 1.636 3.050 3.165 3.771 4.163 4.3503 2.101 2.188 3.241 3.388 4.244 4.390 4.3863.5 3.577 3.898 4.087 4.299 4.541 4.854 4.9054 4.669 4.826 4.904 5.035 5.109 5.530 5.716

Table 4.4The viscosity coefficient s (10�3 Pa s) for a walking knee.

ch/cl (mg/ml) 0.05 0.1 0.15 0.2 0.25 0.3 0.35

1 0.685 0.777 1.120 1.416 1.924 2.273 2.4651.5 0.921 1.944 1.389 1.448 2.037 2.428 2.5832 0.968 1.256 1.830 1.880 2.285 2.527 2.9722.5 1.183 1.595 2.155 2.265 2.465 2.627 3.1523 1.350 1.851 2.547 3.189 3.939 4.279 4.3533.5 2.521 2.611 2.636 3.245 4.086 4.352 4.4564 2.655 2.837 2.907 3.535 4.222 4.411 4.484

V. Chiroiu et al. / Mechanics Research Communications 37 (2010) 246–255 253

parameters and a good verification of all governing equations(Fig. 4.1).

The experimental data Vexpj ¼ fwðrj; ch; clÞ; j ¼ 1;2; . . . ;mg, are

furnished by the caption technique of the motion for walking nor-

mal knees (Anybody Technology) (Blewis et al., 2007; Bhusari,2007; Plaas et al., 2006; Smith and Ghosh, 1987; Hills, 2000; Cookeet al., 1978; Ogston and Stanier, 1953; Caygill and West, 1969; Jayet al., 2007).

The first results of the genetic algorithm provide the values ofthe viscosity coefficients. Tables 4.1 and 4.2 present the viscositycoefficients l and s (10�3 Pa s) for an immobilized normal humanknee and various concentrations ch and cl.

From Tables 4.3 and 4.4, we see that the viscosity coefficients land m are increased by 0.083 ± 0.027 (10�3 Pa s) when compared toan immobilized knee (Tables 4.1 and 4.2).

In comparison with the results reported by Cooke et al. (1978),we can affirm that the present results support the observation ofmajor physical phenomena observed in the behavior of the syno-vial fluid in the human knee.

A new result reveals an interesting phenomenon, namely a de-crease of l with respect to the synovial fluid volume. This result ispresented in Fig. 4.2 for ch = 1, cl = 0.5 for a walking knee. For thecase of a walking knee, Fig. 4.3 depicts the variation of s with re-spect to the synovial fluid volume for ch = 1 and cl = 0.5. It is ob-served a decrease of s for the synovial fluid volume 6 0.138 andthen an increase of s with respect to the synovial fluid volume. Thiseffect is similar to the Fahraesus and Lindquist effect for the bloodmotion in small vessels (Munteanu et al., 2006).

5. Discussions and conclusions

Two micromorphic viscosity coefficients l and s are introducedas functions of the hyaluronan and PRG4/lubricin concentrationsch, cl and cs. Experimentally, the available data are depicted onlyfor l.

We compare now the results obtained for the direct and inverseapproaches, as well as the influence of the modelling assumptionson the obtained results. In the direct approach, the velocityw=w0 ¼ ~vzðrÞ=~vz0 and the gyration components mzr and mrz aredetermined for approximate values l = 3.177, s = 2.271. The esti-mate l = 3.177 was chosen from experimental measurements,without having any relation with ch and cl. The estimate s = 2.271was corrected and re-corrected from the experimental time depen-dent stress Dt12 displayed in Fig. 3.4. In Fig. 3.2 the fixed parametersare cs = 0.1, cl = 0.2, and three values are taken for cl, namely ch = 1,2.5 and 3.5. From Tables 4.3 and 4.4, we depict l = 2.416, 3.165 and4.299 and s = 1.416, 2.265 and 3.245, for the corresponding valuesof ch = 1, 2.5 and 3.5. Therefore, we can conclude that the experi-mental values l and s are close to the theoretical valuesl = 3.165 and s = 2.265, calculated for ch = 2.5. The same analysisis made for Fig. 3.3, for which the same l = 3.177, s = 2.271 are ta-ken. The fixed parameters are cs = 0.1, ch = 2.5, and three values aretaken for cl, i.e. cl = 0.05, 0.2 and 0.35, respectively. From Tables 4.3and 4.4, we depict l = 1.223, 3.165 and 4.350 and s = 1.183, 2.265and 3.152, corresponding to cl = 0.05, 0.2 and 0.35, respectively.Hence the same result as before is obtained, cl = 0.2.

It is clear from the presented tables that the estimation of theviscosity coefficients with respect to the concentration parameters

Fig. 4.2. The variation of the viscosity coefficient l with respect to the synovial fluid volume.

Fig. 4.3. The variation of the viscosity coefficient s with respect to the synovial fluid volume.

254 V. Chiroiu et al. / Mechanics Research Communications 37 (2010) 246–255

ch, cl and cs, are very useful for determining the right values for land s. The sensitivity of the solutions with respect to ch and cl isthe key for the rheopexy of the synovial fluid, namely the synovialfluid displays an increase in the stresses with respect to time dur-ing steady shear, and a decrease in viscosity with respect toincreasing shear rate.

In conclusion, the theory of micromorphic fluids supports theobservation of major physical phenomena noticed in the behaviorof the synovial fluid in the human knee (Cooke et al., 1978). Addi-tionaly, a new result reveals an interesting phenomenon, namely adecrease in the values of l with respect to the synovial fluid vol-

ume (Fig. 4.2) for ch = 1 and cl = 0.5, in the case of a walking knee.The variation of s with respect to the synovial fluid volume is dis-played in Fig. 4.3. It is observed a decrease in the values of s for thesynovial fluid volume 6 0.138 and then an increase in the values ofs with respect to the synovial fluid volume. This effect is similar tothe Fahraesus and Lindquist effect for the blood motion in smallvessels.

We believe that the micromorphic theory highlights the under-standing of the viscous properties of the synovial fluid necessary inthe generation of synovial microflow in microelectromechanicalsystems (MEMS) specialized for joints analyses and medical

V. Chiroiu et al. / Mechanics Research Communications 37 (2010) 246–255 255

applications. We introduce two micromorphic viscosity coeffi-cients as functions of the HA and PRG4/lubricin concentrationsand we show that a genetic algorithm based on experimental datacan reconstruct them.

Acknowledgments

The authors acknowledge the financial support received fromthe National University Research Council CNCSIS, Romania, ProjectPNII nr. 106/2007, code 247/2007.

References

Ateshian, G.A., 1993. A B-spline least-squares surface-fitting method for articularsurfaces of diarthrodial joints. Journal of Biomechanical Engineering 115, 366–373.

Ateshian, G.A., Soslowsky, L.J., Mow, V.C., 1991. Quantitation of articular surfacetopography and cartilage thickness in knee joints usingstrereophotogrammetry. Journal of Biomechanics 24, 761–776.

Ateshian, G.A., Soltz, M.A., Mauck, R.L., Basalo, I.M., Hung, C.T., Lai, W.M., 2003. Therole of osmotic pressure and tension-compression nonlinearity in the frictionalresponse of articular cartilage. Transport in Porous Media 50, 5–33.

Balazs, E., 1982. The physical properties of synovial fluid and the special role ofhyaluronic acid. In: Helfet, A. (Ed.), Disorders of the Knee, second ed. JBLippincott, Philadelphia.

Bansil, R., Stanley, E., LaMont, J.T., 1995. Mucin biophysics. Annual Review ofPhysiology 57, 635–657.

Bhusari, A, 2007. Computational and experimental investigation of the role ofhyaluronic acid–protein interactions in the theology of synovial fluid. Masterthesis, Texas Tech University.

Blewis, M.E., Nugent-Derfus, G.E., Schmidt, T.A., Schumacher, B.L., Sah, R.L., 2007. Amodel of synovial fluid lubricant composition in normal and injured joints.European Cells and Materials 13, 26–39.

Caygill, J.C., West, G.H., 1969. The rheological behaviours of synovial fluid and itspossible relation to joint lubrication. Medical and Biological Engineering andComputing 7 (5), 507–516.

Chiroiu, V., Chiroiu, C., 2003. Inverse Problems. Academiei, Bucharest.Cooke, A., Dowson, D., Wright, V., 1978. The rheology of synovial fluid and some

potential synthetic lubricants for degenerate synovial joints. Engineering inMedicine 7 (2), 66–72.

Cosserat, E., Cosserat, F., 1909. Theorie des corps deformables. Hermann et Fils,Paris.

Davies, D.V., 1946. Synovial membrane and synovial fluid of joints. Lancet 248, 815–822.

Dowson, D., 1990. Bio-tribology of natural and replacement synovial joints. In:Ratcliffe, A., Mow, V., Woo, S. (Eds.), Biomechanics of Diathrodial Joints, vol. 2.Springer-Verlag, New York.

Eckstein, F., Winzheimer, M., Hohe, J., Englmeier, K.H., Reiser, M., 2001.Interindividual variability and correlation among morphological parametersof knee joint cartilage plates: analysis with three-dimensional MR imaging.Osteoarthritis Cartilage 9, 101–111.

Eringen, A.C., 1970a. Balance laws of micromorphic mechanics. InternationalJournal of Engineering Science 8, 819–828.

Eringen, A.C., 1970b. Mechanics of micropolar continua. In: Abir, David (Ed.),Contributions to Mechanics. Pergamon Press, pp. 23–40.

Eringen, A.C., 1998. Microcontinuum Field Theories, II Fluent Media. Springer.

Forster, H., Fisher, J., 1996. The influence of loading time and lubricant on thefriction of articular cartilage. Proceedings of the Institution of MechanicalEngineers 210, 109–119.

Hills, B.A., 2000. Boundary lubrication in vivo. Proceedings of the Institution ofMechanical Engineers [H] 214, 83–94.

Jay, G.D., 2004. Lubricin and surfacing of articular joints. Current Opinion inOrthopaedics 15, 355–359.

Jay, G.D., Torres, J.R., Warman, M.L., Laderer, M.C., Breuer, K.S., 2007. The role oflubricin in the mechanical behavior of synovial fluid. PNAS 104 (15), 6194–6199.

Mazzucco, D., McKinley, G., Scott, R.D., Spector, M., 2002. Rheology of joint fluid intotal knee arthroplasty patients. Journal of Orthopaedic Research 20, 1157–1163.

Mazzucco, D., Scott, R., Spector, M., 2004. Composition of joint fluid in patientsundergoing total knee replacement and revision arthroplasty: correlation withflow properties. Biomaterials 25, 4433–4445.

Mihailescu, M., Chiroiu, V., 2005. Advanced Mechanics on Shells and IntelligentStructures. Academiei, Bucharest.

Morrell, K.C., Hodge, W.A., Krebs, D.E., Mann, R.W., 2005. Corroboration of in vivocartilage pressures with implications for synovial joint tribology andosteoarthritis causation. Proceedings of the National Academy of Sciences ofthe USA 102, 14819–14824.

Mos�negut�u, V., 2008. On the dynamic systems with friction with applications tovibration and damping control. PhD thesis, Romanian Academy.

Munteanu, L., Donescu, S�t., Chiroiu, V., 2006. An inverse problem for the motion ofblood in small vessels. Physiological Measurement 27 (9), 865–880.

Oates, K.M.N., Krause, W.E., Jones, R.L., Colby, R.H., 2006. Rheopexy of synovial fluidand protein aggregation. Journal of the Royal Society, Interface 3 (6), 167–174.

Ogston, A.G., Stanier, J.E., 1953. The physiological function of hyaluronic acid insynovial fluid, viscous, elastic and lubricant properties. Journal of Physiology119 (2–3), 244–252.

Plaas, A., Chekerov, I., Zheng, Y., Schmidt, T., Sah, R., Carter, J., Sandy, J., 2006.Disulfide-bonded multimers of lubricin (LGP-1, PRG4) glycovariants incartilage, synovium and synovial fluid. Transactions Orthopaedic ResearchSociety 52, 1422.

Radin, E.L., Swann, D.A., Weisser, P.A., 1970. Separation of a hyaluronate-freelubricating fraction from synovial fluid. Nature 228, 377–378.

Rhee, D.K. et al., 2005. The secreted glycoprotein lubricin protects cartilage surfacesand inhibits synovial cell overgrowth. Journal of Clinical Investigation 115 (3),622–631.

Ropes, M.W., Rossmeisl, E.C., Bauer, W., 1940. The origin and nature of normalhuman synovial fluid. Journal of Clinical Investigation 19, 795–799.

Schmid, T., Lindley, K., Su, J., Soloveychik, V., Block, J., Kuettner, K., Schumacher, B.,2001. Superficial zone protein (SZP) is an abundant glycoprotein in humansynovial fluid and serum. Transactions Orthopaedic Research Society 26, 82.

Schwarz, I.M., Hills, B.A., 1998. Surface-active phospholipids as the lubricatingcomponent of lubricin. British Journal of Rheumatology 37, 21–26.

Simkin, P.A., 2000. Friction and lubrification in synoivial joints. The Journal ofRhematology 27, 567–568.

Smith, M.M., Ghosh, P., 1987. The synthesis of hyaluronic acid by human synovialfibroblasts is influenced by the nature of the hyaluronate in the extracellularenvironment. Rheumatology International 7, 113–122.

Swann, D.A. et al., 1981. The molecular structure of lubricating glycoprotein-I, theboundary lubricant for articular cartilage. Journal of Biological Chemistry 256,5921–5925.

Weissman, B., Meyer, K., 1954. The structure of hyalobiuronic acid and of hyaluronicacid from umbilical cord. Journal of the American Chemical Society 76, 1753–1757.

Zappone, B., Ruths, M., Greene, G.W., Jay, G.D., Israelachvili, J.N., 2007. Adsorption,lubrication, and wear of lubricin on model surfaces: polymer brush-likebehavior of a glycoprotein. Biophysical Journal 92, 1693–1708.